A uniform boundedness theorem

Abstract

A variation on the uniform boundedness theorem [2, p. 105] of functional analysis is obtained by replacing pointwise boundedness by a slightly stronger type of boundedness. This makes it possible to remove the category requirements from the set over which it is desired to have a collection of functions uniformly bounded. The concept of boundedness to be utilized is called boundedness on Cauchy sequences (see Definition 1). When a Cauchy sequence converges in the domain of a family of continuous functions, boundedness on a Cauchy sequence is equivalent to pointwise boundedness of the family at the limit point of the point of the sequence. When the domain of the continuous functions is a metric space, boundedness on the collection on all Cauchy sequences is equivalent to pointwise boundedness of the continuous extensions on the completion of the domain. The interesting case is when the domain is not metrizable because boundedness on Cauchy sequences does not necessarily imply pointwise boundedness of the extended functions on the completion or even on the sequential completion of the domain. The main result of the paper is given in Theorem 1 which is followed by six corollaries. The sixth corollary is especially interesting in stating that the closure of an infrabarrelled space in its second adjoint space is a barreled space. The theorem on uniform boundedness immediately suggests an absorption theorem which is Theorem 2 of the paper. Theorem 1 does not follow as a corollary of Theorem 2 as is the case of the classical uniform boundedness theorem [2, p. 105]. DEFINITION 1. A family H of continuous linear transformations with common domain a linear topological space E and ranges in a locally convex space F is bounded on a Cauchy sequence {Xn } in E if for every neighborhood V of the origin in F there exists a positive real number r with the property that for each f in H there is an Nf such that f (xn) is in r V for all n > Nf. The following theorem is stated in the context of the definition.